Discrete Mathematics in Computer Science B8. Functions Malte Helmert, Gabriele R¨ oger University of Basel October 19, 2020 Malte Helmert, Gabriele R¨ oger (University of Basel) Discrete Mathematics in Computer Science October 19, 2020 1 / 34
Discrete Mathematics in Computer Science October 19, 2020 — B8. Functions B8.1 Partial and Total Functions B8.2 Operations on Partial Functions B8.3 Properties of Functions Malte Helmert, Gabriele R¨ oger (University of Basel) Discrete Mathematics in Computer Science October 19, 2020 2 / 34
B8. Functions Partial and Total Functions B8.1 Partial and Total Functions Malte Helmert, Gabriele R¨ oger (University of Basel) Discrete Mathematics in Computer Science October 19, 2020 3 / 34
B8. Functions Partial and Total Functions Important Building Blocks of Discrete Mathematics Important building blocks: ◮ sets ◮ relations ◮ functions In principle, functions are just a special kind of relations: ◮ f : N 0 → N 0 with f ( x ) = x 2 ◮ relation R over N 0 with R = { ( x , y ) | x , y ∈ N 0 and y = x 2 } . Malte Helmert, Gabriele R¨ oger (University of Basel) Discrete Mathematics in Computer Science October 19, 2020 4 / 34
B8. Functions Partial and Total Functions Functional Relations Definition A binary relation R over sets A and B is functional if for every a ∈ A there is at most one b ∈ B with ( a , b ) ∈ R . a 1 a 1 2 2 b b B B A c 3 A c 3 4 4 d d e e functional not functional Malte Helmert, Gabriele R¨ oger (University of Basel) Discrete Mathematics in Computer Science October 19, 2020 5 / 34
B8. Functions Partial and Total Functions Functions – Examples ◮ f : N 0 → N 0 with f ( x ) = x 2 + 1 ◮ abs : Z → N 0 with � x if x ≥ 0 abs ( x ) = − x otherwise ◮ distance : R 2 × R 2 → R with ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 � distance (( x 1 , y 1 ) , ( x 2 , y 2 )) = Malte Helmert, Gabriele R¨ oger (University of Basel) Discrete Mathematics in Computer Science October 19, 2020 6 / 34
B8. Functions Partial and Total Functions Partial Function – Example Partial function r : Z × Z � Q with � n if d � = 0 d r ( n , d ) = undefined otherwise Malte Helmert, Gabriele R¨ oger (University of Basel) Discrete Mathematics in Computer Science October 19, 2020 7 / 34
B8. Functions Partial and Total Functions Partial Functions Definition (Partial function) A partial function f from set A to set B (written f : A � B ) is given by a functional relation G over A and B . Relation G is called the graph of f . We write f ( x ) = y for ( x , y ) ∈ G and say y is the image of x under f . If there is no y ∈ B with ( x , y ) ∈ G , then f ( x ) is undefined. Partial function r : Z × Z � Q with � n if d � = 0 d r ( n , d ) = undefined otherwise d ) | n ∈ Z , d ∈ Z \ { 0 }} ⊆ Z 2 × Q . has graph { (( n , d ) , n Malte Helmert, Gabriele R¨ oger (University of Basel) Discrete Mathematics in Computer Science October 19, 2020 8 / 34
B8. Functions Partial and Total Functions Domain (of Definition), Codomain, Image Definition (domain of definition, codomain, image) Let f : A � B be a partial function. Set A is called the domain of f , set B is its codomain. The domain of definition of f is the set dom( f ) = { x ∈ A | there is a y ∈ B with f ( x ) = y } . The image (or range) of f is the set img( f ) = { y | there is an x ∈ A with f ( x ) = y } . f : { a , b , c , d , e } � { 1 , 2 , 3 , 4 } a 1 f ( a ) = 4 , f ( b ) = 2 , f ( c ) = 1 , f ( e ) = 4 b 2 domain { a , b , c , d , e } B A c 3 codomain { 1 , 2 , 3 , 4 } d 4 domain of definition dom( f ) = { a , b , c , e } e image img( f ) = { 1 , 2 , 4 } Malte Helmert, Gabriele R¨ oger (University of Basel) Discrete Mathematics in Computer Science October 19, 2020 9 / 34
B8. Functions Partial and Total Functions Preimage The preimage contains all elements of the domain that are mapped to given elements of the codomain. Definition (Preimage) Let f : A � B be a partial function and let Y ⊆ B . The preimage of Y under f is the set f − 1 [ Y ] = { x ∈ A | f ( x ) ∈ Y } . f − 1 [ { 1 } ] = a 1 f − 1 [ { 3 } ] = b 2 B A c 3 f − 1 [ { 4 } ] = 4 d f − 1 [ { 1 , 2 } ] = e Malte Helmert, Gabriele R¨ oger (University of Basel) Discrete Mathematics in Computer Science October 19, 2020 10 / 34
B8. Functions Partial and Total Functions Total Functions Definition (Total function) A (total) function f : A → B from set A to set B is a partial function from A to B such that f ( x ) is defined for all x ∈ A . → no difference between the domain and the domain of definition a 1 b 2 B A c 3 d 4 e Malte Helmert, Gabriele R¨ oger (University of Basel) Discrete Mathematics in Computer Science October 19, 2020 11 / 34
B8. Functions Partial and Total Functions Specifying a Function Some common ways of specifying a function: ◮ Listing the mapping explicitly, e. g. f ( a ) = 4 , f ( b ) = 2 , f ( c ) = 1 , f ( e ) = 4 or f = { a �→ 4 , b �→ 2 , c �→ 1 , e �→ 4 } ◮ By a formula, e. g. f ( x ) = x 2 + 1 ◮ By recurrence, e. g. 0! = 1 and n ! = n ( n − 1)! for n > 0 ◮ In terms of other functions, e. g. inverse, composition Malte Helmert, Gabriele R¨ oger (University of Basel) Discrete Mathematics in Computer Science October 19, 2020 12 / 34
B8. Functions Partial and Total Functions Relationship to Functions in Programming def factorial(n): if n == 0: return 1 else: return n * factorial(n-1) → Relationship between recursion and recurrence Malte Helmert, Gabriele R¨ oger (University of Basel) Discrete Mathematics in Computer Science October 19, 2020 13 / 34
B8. Functions Partial and Total Functions Relationship to Functions in Programming def foo(n): value = ... while <some condition>: ... value = ... return value → Does possibly not terminate on all inputs. → Value is undefined for such inputs. → Theoretical computer science: partial function Malte Helmert, Gabriele R¨ oger (University of Basel) Discrete Mathematics in Computer Science October 19, 2020 14 / 34
B8. Functions Partial and Total Functions Relationship to Functions in Programming import random counter = 0 def bar(n): print("Hi! I got input", n) global counter counter += 1 return random.choice([1,2,n]) → Functions in programming don’t always compute mathematical functions (except purely functional languages ). → In addition, not all mathematical functions are computable. Malte Helmert, Gabriele R¨ oger (University of Basel) Discrete Mathematics in Computer Science October 19, 2020 15 / 34
B8. Functions Operations on Partial Functions B8.2 Operations on Partial Functions Malte Helmert, Gabriele R¨ oger (University of Basel) Discrete Mathematics in Computer Science October 19, 2020 16 / 34
B8. Functions Operations on Partial Functions Restrictions and Extensions Definition (restriction and extension) Let f : A � B be a partial function and let X ⊆ A . The restriction of f to X is the partial function f | X : X � B with f | X ( x ) = f ( x ) for all x ∈ X . A function f ′ : A ′ � B is called an extension of f if A ⊆ A ′ and f ′ | A = f . The restriction of f to its domain of definition is a total function. What’s the graph of the restriction? What’s the restriction of f to its domain? Malte Helmert, Gabriele R¨ oger (University of Basel) Discrete Mathematics in Computer Science October 19, 2020 17 / 34
B8. Functions Operations on Partial Functions Function Composition Definition (Composition of partial functions) Let f : A � B and g : B � C be partial functions. The composition of f and g is g ◦ f : A � C with g ( f ( x )) if f is defined for x and ( g ◦ f )( x ) = g is defined for f ( x ) undefined otherwise Corresponds to relation composition of the graphs. If f and g are functions, their composition is a function. Example: with f ( x ) = x 2 f : N 0 → N 0 g : N 0 → N 0 with g ( x ) = x + 3 ( g ◦ f )( x ) = Malte Helmert, Gabriele R¨ oger (University of Basel) Discrete Mathematics in Computer Science October 19, 2020 18 / 34
B8. Functions Operations on Partial Functions Properties of Function Composition Function composition is ◮ not commutative: ◮ f : N 0 → N 0 with f ( x ) = x 2 ◮ g : N 0 → N 0 with g ( x ) = x + 3 ◮ ( g ◦ f )( x ) = x 2 + 3 ◮ ( f ◦ g )( x ) = ( x + 3) 2 ◮ associative, i. e. h ◦ ( g ◦ f ) = ( h ◦ g ) ◦ f → analogous to associativity of relation composition Malte Helmert, Gabriele R¨ oger (University of Basel) Discrete Mathematics in Computer Science October 19, 2020 19 / 34
B8. Functions Operations on Partial Functions Function Composition in Programming We implicitly compose functions all the time. . . def foo(n): ... x = somefunction(n) y = someotherfunction(x) ... Many languages also allow explicit composition of functions, e. g. in Haskell: incr x = x + 1 square x = x * x squareplusone = incr . square Malte Helmert, Gabriele R¨ oger (University of Basel) Discrete Mathematics in Computer Science October 19, 2020 20 / 34
B8. Functions Properties of Functions B8.3 Properties of Functions Malte Helmert, Gabriele R¨ oger (University of Basel) Discrete Mathematics in Computer Science October 19, 2020 21 / 34
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